i)
\(\mathcal {A}\)  is the heart of a bounded \(t\) -structure, in particular it is an Abelian subcategory of \(\mathcal {C}\) ;
ii)
\(Z\colon \Gamma \rightarrow {C}\)  is a linear map, called the central charge;
iii)
For any \(0\ne E\in \mathcal {A}\) , \(Z(E)=\rho (E) e^{I\pi \phi (E)}\)  where \(\rho (E)>0\) 
and \(0<\phi (E)\le 1\) ; in other words, \(Z(E)\)  is contained in \({H}_B={H}\cup (-\infty ,0)\) ;
iv)
(Harder-Narasimhan property) Every \(0\ne E\in \mathcal {A}\)  admits a finite filtration \(0 \subset E_0\subset E_1 \cdots \subset E_n =E\)  by objects \(E_i\)  in \(\mathcal {A}\) , such that each factor \(F_iE_i/E_{i-1}\)  is \(\sigma \) -semistable
and \(\phi (F_1)>\phi (F_2)>\ldots > \phi (F_n)\) ;
v)
(Support property) There exists a quadratic form \(Q\)  on \(\Gamma \otimes {R}\)  such that the kernel of \(Z\)  in
\(\Gamma \otimes {R}\)  is negative definite with respect to \(Q\) , and moreover, for any
\(\sigma \) -semistable object \(E\in \mathcal {A}\) , \(Q(\gamma (E))\ge 0\) . Equivalently [1]},
there exists a non-negative constant \(C\)  such that, for all \(\sigma \) -semistable object \(E\in \mathcal {A}\) ,
\(\Vert  \gamma (E) \Vert  \le C \, |Z(E)|\) 
where \(\Vert  \cdot \Vert \)  is a fixed Euclidean norm on \(\Gamma \otimes {R}\) .

